Well I also have a very similar story. Being a natural lefty, I started off playing a common right handed guitar in an upside down orientation. And yes I did not make any changes in the strings. So basically a down strum for me meant EBGDAE rather than the usual EADGBE. So I started strumming in the opposite direction. For e.g. a down-down-down-down pattern for me was up-up-up-up. And naturally all the chord shapes for me were in upside down positions. I continued with this practice for about 6-7 months during a period in which I matured from an absolute newbie to an advanced beginner kind of a player.
This has been my story till now but now I am seriously contemplating changing over to a usual right handed guitar player. Over a period of time with increasing guitar maturity, I have somehow started feeling that this upside-down method of playing is presenting more obstacles than advantages. While it feels natural for me to play it this way and that's why I started off this way as a newbie , there have always been adjustments that I had to make while learning. For e.g. most of the online guitar lessons worth following are primarily for right handers with sometimes a fleeting mention for left handers. The chord shapes are all reversed and this starts getting troublesome as more complex ones start coming in. For example, I could easily manage with the mirror images of major A,D,E,G,C chords and many minor ones as well but then when I came across chords like B7, C7 Gsus, it was simply beyond me to achieve the mirror images of those shapes with my fretting hand. In some cases I could barely manage with my pinky finger on the thickest string but that is not ideal. I think these chord shapes have been shaped for a specific hand and using the other hand for them is just a makeshift arrangement which would start falling apart as things get more involved.
I also feel that strumming "upwards" has an inherently different sound from strumming down even when you are hitting the same sequence of string. Maybe it the gravity assisting the downward motion or the physiological structure of our hand muscles, I don't know but I have a feeling that the sound is different and this difference becomes more pronounced as you advance with your playing skills. And finally the clincher which made me make up my mind to change my "playing handedness" came when I was exploring electric guitars contemplating buying one sometime in future. As I saw, the handedness becomes even more pronounced with electric guitars with all their volume knobs and controls placed in an awkward position for upside down playing. Probably there's an option of dismantling and restringing the whole guitar but that still means that I have to relearn my chord shapes and strumming patterns all over again. So if I have to undergo that change, I'd prefer to go a step further and change my playing and fretting hand as that way I would not be restricted by customized left handed guitars and would have a wide variety of guitars to choose from.
With all that said, changing my playing handedness would still mean that I have to start again as a newbie from scratch. Get all those finger calluses yet again, relearn all the chord shapes, chord changes, strumming patterns. Feels like flushing down the sincere effort that I had put in for the last 6-7 months down the drain. And most painfully, going back from someone who knows to play half-decent guitar to someone who does not.
But I guess it is better to be late than never. For now it's a matter of relearning what I learnt in the past 6-7 months but perhaps it is better than putting in years and then reaching a point of no return.
That was my story but let me emphasize that one must play the way they like the best. After all the ultimate intent is to derive satisfaction from it and it isn't just a mechanical exercise which can be done only in a specified way.